Sen’s Liberal paradox describes a conflict between weak Pareto, minimal liberalism, and either transitivity or a best element over a domain of individual preferences. This paper examines variants of that paradox with varying amounts of unconditional preferences. We define a notion of unconditional preferences under which, in the absence of Pareto, there can be no cycles. We then define a stronger condition, that makes an individual’s preferences for her own private attributes independent of all other attributes. Under this assumption, there can be no cycles with or without Pareto. We also show there exists a social decision function satisfying those conditions. We then determine the probability of a cycle assuming a much weaker independence condition that does not restrict the domain. This probability converges to one as the number of non-private attributes within the social states increases. Finally, we use simulations to determine the probability that liberalism and Pareto conflict with best elements, maximal elements, and transitivity separately.
Dougherty, K.L., Edward, J. The effect of unconditional preferences on Sen’s paradox. Theory Decis 93, 427–447 (2022). https://doi.org/10.1007/s11238-021-09863-8
